Revenue Gaps for Static and Dynamic Posted Pricing of Homogeneous Goods

We consider the problem of maximizing the expected revenue from selling $k$ homogeneous goods to $n$ unit-demand buyers who arrive sequentially with independent and identically distributed valuations. In this setting the optimal posted prices are dynamic in the sense that they depend on the remaining numbers of goods and buyers, and may be difficult to implement due to technical or regulatory restrictions. We thus investigate how much revenue is lost when a single static price is used instead for all buyers and goods, and prove upper bounds on the ratio between the maximum revenue from dynamic prices and that from static prices. These bounds turn out to be tight for all values of $k$ and $n$ and vary depending on a regularity property of the underlying distribution. For general distributions we obtain a ratio of $2-k/n$, for regular distributions a ratio that increases in $n$ and is bounded from above by $1/(1-k^k/(e^{k}k!))$, which is of order $1/(1-1/sqrt{k})$. The lower bounds hold for the revenue gap between dynamic and static prices. The upper bounds are obtained via an ex-ante relaxation of the revenue maximization problem, as a consequence the tight bounds of $2-k/n$ in the general case and of order $1/(1-1/sqrt{k})$ in the regular case apply also to the potentially larger revenue gap between the optimal incentive-compatible mechanism and the optimal static price.